Given samples \(X_1,X_2,\ldots \in \mathbb {R}^k\)  drawn from some unknown smooth and compact subset \(M\subset \mathbb {R}^n\) , an inference problem of general interest is to estimate the intrinsic distances \(\mathrm {dist}_M(X_p,X_q)\)  in \(M\) , the shortest lengths of paths in \(M\).
This general problem has motivations throughout data analysis.
One motivation comes from manifold learning.
In fact, consistently estimating intrinsic distances in \(M\)  is non-parametric manifold learning (c.f. [1]}): one is learning a sequence of finite metric spaces which converge in a suitable sense [2]}.
Another motivation comes from network inference under a natural generative model in terms of a latent space \(M\)  of nodes; good estimates of intrinsic distances in \(M\)  can yield good estimates of the generating distribution (eg. [3]}.)
Another motivation comes from dimensionality reduction, where \(\dim \,M\ll k\) ; good estimates of intrinsic distances in \(M\)  can yield good embeddings of \(X_1,X_2,\ldots \)  into \(M\)  up to isometry [4]}, [5]}, [6]}.
Another motivation comes from regression, where knowledge of intrinsic manifold distances yield more efficient estimators [7]}.
Yet another motivation comes from classification, where the replacement of \(\mathrm {dist}_M(X_p,X_q)\)  in place of Euclidean distances \(\Vert X_p-X_q\Vert \)  in various algorithms often improves performance (eg. [8]}).
